Subgroups — C₁₁×#537 #10623398

Order 704.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#41	C₂	1	2	yes	maximal, complemented
2	C₁₁×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₁₁×#121	C₂	1	4	yes	maximal, complemented
4	C₁₁×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#41	—	2	8	no	—
7	C₂×Q₈×C₁₁	C₂²	1	2	yes	—
8	C₂×D₄×C₁₁	C₂²	1	4	yes	—
9	C₂×D₄×C₁₁	C₂²	1	2	yes	—
10	C₂×D₄×C₁₁	C₂²	1	4	yes	—
11	C₂²×C₄×C₁₁	C₂²	1	4	yes	—
12	C₂²×C₄×C₁₁	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
15	C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
16	C₁₁×C₂²⋊C₄	C₂²	1	8	yes	—
17	C₂³×C₁₁	C₂³	1	4	yes	—
18	C₂³×C₁₁	C₂³	1	1	yes	characteristic
19	Q₈×C₁₁	—	2	4	no	—
20	D₄×C₁₁	—	2	8	no	—
21	D₄×C₁₁	—	2	4	no	—
22	D₄×C₁₁	—	4	4	no	—
23	C₂×C₄×C₁₁	—	2	8	no	—
24	C₂×C₄×C₁₁	—	4	2	no	—
25	C₂×C₄×C₁₁	D₄	1	4	yes	complemented
26	C₂×C₄×C₁₁	C₂³	1	4	yes	—
27	C₂×C₄×C₁₁	C₂³	1	4	yes	—
28	C₂×C₄×C₁₁	C₂³	1	2	yes	—
29	#537	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₁₁	—	2	1	no	—
31	C₂²×C₁₁	—	2	1	no	—
32	C₂²×C₁₁	—	2	4	no	—
33	C₂²×C₁₁	—	2	4	no	—
34	C₂²×C₁₁	—	2	4	no	—
35	C₂²×C₁₁	C₂×D₄	1	2	yes	—
36	C₂²×C₁₁	C₂⁴	1	1	yes	characteristic
37	C₄×C₁₁	—	2	4	no	—
38	C₄×C₁₁	—	2	4	no	—
39	C₄×C₁₁	C₂×D₄	1	4	yes	—
40	C₂×#41	C₂×C₁₁	1	2	yes	complemented
41	C₂²⋊Q₈	C₂×C₁₁	1	4	yes	complemented
42	#121	C₂×C₁₁	1	4	yes	complemented
43	#121	C₂×C₁₁	1	4	yes	complemented
44	C₂×C₄⋊C₄	C₂×C₁₁	1	1	yes	characteristic, complemented
45	C₂×C₁₁	—	2	2	no	—
46	C₂×C₁₁	—	4	4	no	—
47	C₂×C₁₁	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
49	C₂×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
50	#41	—	2	8	no	—
51	C₂×Q₈	C₂²×C₁₁	1	2	yes	—
52	C₂×D₄	C₂²×C₁₁	1	4	yes	—
53	C₂×D₄	C₂²×C₁₁	1	2	yes	—
54	C₂×D₄	C₂²×C₁₁	1	4	yes	—
55	C₂²×C₄	C₂²×C₁₁	1	4	yes	—
56	C₂²×C₄	C₂²×C₁₁	1	2	yes	—
57	C₂²×C₄	C₂²×C₁₁	1	1	yes	characteristic
58	C₄⋊C₄	C₂²×C₁₁	1	4	yes	—
59	C₄⋊C₄	C₂²×C₁₁	1	4	yes	—
60	C₂²⋊C₄	C₂²×C₁₁	1	8	yes	—
61	C₁₁	#537	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³	C₂³×C₁₁	1	4	yes	—
63	C₂³	C₂³×C₁₁	1	1	yes	characteristic
64	Q₈	—	2	4	no	—
65	D₄	—	2	8	no	—
66	D₄	—	2	4	no	—
67	D₄	—	4	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	D₄×C₁₁	1	4	yes	complemented
71	C₂×C₄	C₂³×C₁₁	1	4	yes	—
72	C₂×C₄	C₂³×C₁₁	1	4	yes	—
73	C₂×C₄	C₂³×C₁₁	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×D₄×C₁₁	1	2	yes	—
80	C₂²	C₂⁴×C₁₁	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	2	4	no	—
83	C₄	C₂×D₄×C₁₁	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	4	4	no	—
86	C₂	C₂²×D₄×C₁₁	1	1	yes	characteristic
87	C₂	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
88	C₂	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₁₁×#41	C₂	semidirect
2. C₁₁×C₂²⋊Q₈	C₂	semidirect
3. C₁₁×#121	C₂	semidirect
4. C₁₁×#121	C₂	semidirect
5. C₂×C₁₁×C₄⋊C₄	C₂	semidirect
25. C₂×C₄×C₁₁	D₄	semidirect
29. #537	C₁₁	direct (N × H)
40. C₂×#41	C₂×C₁₁	semidirect
41. C₂²⋊Q₈	C₂×C₁₁	semidirect
42. #121	C₂×C₁₁	semidirect
43. #121	C₂×C₁₁	semidirect
44. C₂×C₄⋊C₄	C₂×C₁₁	semidirect
61. C₁₁	#537	direct (N × H)
70. C₂×C₄	D₄×C₁₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₁×#41	2
C₁₁×C₂²⋊Q₈	2
C₁₁×#121	2
C₂×C₁₁×C₄⋊C₄	2
#537	11

Subgroups — C11×#537 #10623398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#537 #10623398